PHY197F notes

Cosmology: Weighing the Universe

Carl Sagan’s Cosmic Calendar

Carl Sagan was one of the most brilliant science communicators who ever lived. One of the devices he used to help communicate cosmic time scales is his Cosmic Calendar included here for illustration.

[Carl Sagan's cosmic calendar, depicting the timescales on which astrophysical and biological processes relevant to our life on Earth occurred, by compressing the age of the universe into one year. For example: the oldest known life form did not appear on the scene until September, and anatomically modern humans did not appear until 11:54pm on December 31st]

The most amazing thing that this illustrates is how long it took for our Sun to evolve and then our planet and all life on it. Billions and billions of years. As far as we know, the Universe is approximately 13.7 billion years old, plus or minus a few hundred thousand years. And we know this by using Einstein’s famous theory of General Relativity.

Olbers’ Paradox

Have you ever wondered why the night sky is mostly dark, punctuated by starlight? Assume for the moment that the universe is approximately the same in every direction, which is a decent first approximation. So there ought to be stars in every direction that we could point in. But not all stars shine with the same brightness. Indeed, we know that for electromagnetism there is an inverse-square law. Stars that are three times further away are nine times as dim, and so forth. A competing effect is that as you go further away, the surface of the night sky a certain distance away (a sphere) grows with radius like radius-squared. Combining these two facts -- the inverse-square law for star brightness and how spheres grow with radius -- tells us that the night sky should be ablaze with light in every direction. The fact that it isn’t is called Olbers’ Paradox.

Olbers’ conclusion is the correct conclusion if the universe is static, i.e. not moving, and also infinite in size. The solution to Olbers’ Paradox is absolutely mindblowing: our observable universe is actually finite and expanding! So how do astrophysicists know this?

Hubble’s Law

In 1929, Edwin Hubble put together a bunch of data (making use of other physicists’ data, e.g. Slipher) to propose a law saying that the universe is expanding. What led him to propose such a heretical idea? Well, he looked at a spectrographic analysis of starlight. In plain English this means that he looked at the emission spectrum of each star in the data set, and analyzed them all. Why was this so important?

Quantum mechanics tells us the photon energy and frequency are related by $E=hf$. If the photon frequency $f$ gets redshifted compared to what is observed in the atomic rest frame, then by special relativity we know that it is moving away from us and how fast. Hubble found that, on average, the more distant light sources were moving away from us faster. $$ v = H_0\, D \,. $$ This says that the speed $v$ is proportional to the distance $D$ with proportionality constant $H_0$. The constant is known as the Hubble constant.

Physicists realized that the reason why the universe is expanding is that the fabric of spacetime itself is expanding. It can be explained in a similar way to how dots drawn on a balloon that is then blown up move apart from each other faster as the balloon is blown up. Note that, when spacetime expands like this, it does not change the size of gravitationally bound systems like stars with planets or galaxies. It just increases the space in between galaxies and stuff.

[an image denoting, in two frames, the idea of expansion of the fabric of the universe. ]

Causality and Hubble’s Law

Once you have looked at the Hubble equation for awhile, a scary thought appears to you. What if the galaxy was far enough away that the recessional speed became greater than the speed of light? What would this mean? Does it mean that signals are actually being transmitted faster than the speed of light in violation of Einstein’s fundamental speed limit!?

No. You can rest easy. There is another way to interpret Hubble’s Law that gives a much better understanding of how our expanding universe operates. Actually, galaxies which appear to go faster than light are galaxies we cannot see. The light from them simply cannot reach us any more. We call the patch of spacetime containing things with which we could in principle communicate the causal patch. It has a finite size because our universe has lived for only a finite amount of time (about 13.7 billion years). We can only see out to our cosmological horizon, and physics within the horizon is completely consistent with causality. We are simply out of causal contact with all stuff that is outside our cosmological horizon. The principle of causality is safe.

Dark Matter and Dark Energy

Dark matter is defined as matter which doesn’t shine. It gravitates but does not interact with photons. It gravitates just the same as regular matter, so can be weighed the same way. Dark matter might be composed of superparticles, but this is speculative.

Dark energy is a lot more mysterious. It corresponds to having a constant energy density, i.e., a constant amount of energy per unit volume of space. The weirdest thing about dark energy is that it gravitates like it has negative pressure, which seems physically crazy in the sense that we don’t yet know of any substance that behaves like this! Despite decades of work by the world’s smartest physicists, nobody yet has a sensible theory of what dark energy is. People have some partial ideas, but they all have logical holes in them. If you can crack the puzzle of dark energy, then you will surely win the admiration of your colleagues worldwide as well as a Nobel Prize.

Pie Chart of the Universe

Einstein’s equations are very well tested indeed. Weighing the universe using them gives a pretty spectacular result:

[pie chart of the universe showing approximate percentages composing our cosmos. Over 70% is dark energy, about 25% is dark matter, and less than 5% is made of regular particles like hydrogen, helium, stars, heavy elements, and neutrinos.]

Why is this pie chart surprising? Well, it says that 95% of the universe is composed of completely foreign stuff that we have not yet seen in the lab!

These results come from combining multiple techniques: CMBR, Type Ia supernovae, Large Scale Structure, Hubble, gravitational lensing, etc. No single technique is king. It is the combination of data-gathering experiments that makes current understanding of cosmological parameters so precise compared to twenty years ago.

CMBR Experiments

What kinds of apparatus did humans use to weigh the CMBR?

[pictures of the experiments of Penzias and Wilson, COBE, and Planck, alongside all-sky images of the cosmic microwave background radiation obtained using those three pieces of equipment]

Penzias and Wilson discovered the CMBR using a ground-based detector. COBE was the first CMBR satellite, put in orbit in 1990. Its results were incredibly important and extremely influential. It discovered anisotropy in the CMBR at the level of one part in 100,000. John Mather and George Smoot, PIs for the project, won the Nobel Prize.

Other CMBR experiments were also important, including balloon-borne experiments like Boomerang. The Wilkinson Microwave Anisotropy Probe (WMAP, a NASA satellite) produced first results in 2003. WMAP gave us a much more detailed picture of small deviations from the perfect blackbody spectrum. Planck (an ESA satellite) was launched in May 2009. There are many other CMBR experiments as well. At any rate, the age of precision cosmology is here.

Type Ia Supernovae

Figuring out expansion of the universe (à la Hubble, etc.) is not straightforward, because you cannot measure the distance of stars/galaxies independently of their brightness. A very few astronomical objects are sufficiently well understood to be called standard candles. Type Ia supernovae are a class of such objects, which all form from similar-mass white dwarfs. Observing the light curves (the graph of the luminosity versus time) of many exploding Type Ia supernovae across the sky was enough to constrain cosmological parameters already in the late 1990s. Combining the discovery of the CMBR with SNIa information produced the discovery of dark energy. This was earth-shaking for the astronomy and physics communities.

FRW Cosmology

Early astrophysicists established the Cosmological Principle: the fact that the universe looks isotropic (the same in all directions) and homogeneous (the same here as over there). Suppose that we model the universe by assuming that space grows with time t via a simple scale factor that is a function of time, a(t). The people who first wrote down the Einstein equations and solved them were Friedmann, Robertson, and Walker. These cosmologies (spacetimes approximating our universe) are known as FRW universes.

So what do Einstein’s equations say about the evolution of the scale factor $a(t)$ with time? They give the scale factor velocity $\dot{a}(t)$ in terms of the Newton constant $G_N$, the energy density $\rho$, and the curvature $\kappa$: $$ \left({\frac{\dot{a}}{a}}\right)^2 = {\frac{8\pi G_N}{3}}\rho - {\frac{\kappa}{a^2}} \,, $$ So if you know how much energy density there is (i.e., energy per cubic metre of space), and the curvature of space, you can work out how fast the universe is expanding. Alternatively, if you can measure the expansion rate and know the curvature, you can infer the energy density that must have been there to make it happen. This is the basic idea behind why you can weigh the universe in the first place.

Einstein's equations also give the scale factor acceleration $\ddot{a}(t)$ in terms of the Newton constant $G_N$, the energy density $\rho$ and the pressure $p$, $$ {\frac{\ddot{a}}{a}}= - {\frac{4\pi G_N}{3}} \left( \rho +3p \right) \,. $$ This equation says that if you know both the energy density and the pressure, you can work out how much the expansion of the universe is speeding up! These Einstein equations are pretty powerful, aren't they? The Hubble parameter, an important piece of information that cosmologists can measure about our universe, is related to the growing scale factor by $$ H = {\frac{\dot{a}}{a}} \,. $$ Another variable called $\Omega$, the density of everything in the universe combined, obeys $$ \Omega = {\frac{8\pi G_N}{3H^2}}\rho = 1 + {\frac{\kappa}{H^2 a^2}} \,. $$

Now we get to the punch line. The physics upshot of all these mathy-looking equations is that knowing the curvature of space $\kappa$ tells you how to predict the future evolution of the universe! So, curvature really matters. Positive curvature (like a basketball) has $\kappa$=+1; negative curvature (like a Pringle) has $\kappa$=-1; flat has $\kappa$=0. The above equation for the density $\Omega$ says that if the universe is undergoing accelerated expansion, then it will tend to become closer and closer to spatially flat. To the best of our current experimental knowledge, the universe is spatially flat.

[cartoons of surfaces with spatial curvatures which are positive (sphere), negative (saddle) and zero (flat)]

Motivations for Cosmic Inflation

Inflation was invented in 1965 by Zeldovich and independently by Alan Guth in 1980. One of the motivations was the observed isotropy of the CMBR. On the face of it, it is surprising that the CMBR would be so uniform on scales bigger than a couple of degrees on our sky. It was unclear to astrophysicists of the day what mechanism would cause wider regions to have (approximately) the same temperature. Inflation explains this. Another fact that points towards inflation is the fact that our universe has been measured to be (extremely close to) flat. Furthermore, we have seen zero magnetic monopoles so far, but they are objects that are typically predicted in GUTs (grand unified theories) relevant to the very earliest epochs. The solution to each of these problems, and a few more besides, is inflation.

Big Bang Misconceptions

Let me outline and address a few possible misconceptions about the Big Bang. The Big Bang as used by modern cosmologists refers to evolution of the universe, rather than its (speculative) origins. Also, the Big Bang is not an expansion into space: instead, the Big Bang made space. Big Bang nucleosynthesis (BBN), which created hydrogen helium and lithium, does not depend on what theory you used to make sense of the Planck epoch. Astrophysicists have a good handle on how to compute relative abundances of different elements and isotopes. But we honestly don't know if the universe started infinitely small and infinitely hot, or as something completely different-looking. Theoretical cosmologists are working hard on this question and most of what is known today is still very controversial.

So what is the edge of our knowledge comfort zone right now? We are experimentally confident about the history of the universe, as we will present next week, starting from baryogenesis and electroweak symmetry breaking. The LHC will be able to help clarify these physics issues significantly. The parts of the story about inflation (the universe’s huge early growth spurt) and the Planck epoch -- the really early universe -- are still considered speculative. Astrophysicists and physicists will continue looking for indirect ways to test ideas experimentally, but we may well end up fundamentally limited in our ability to know about the universe. Finding those limits is still very much a work in progress too.

BBN and Prediction of CMBR

Einstein’s static universe was one early contender for a theory of the entire cosmos. Another contender was Lemaître and Gamow’s Big Bang Nucleosynthesis (BBN) model. Their associates Alpher and Herman predicted the cosmic microwave background radiation (CMBR) from this theory. The CMBR was discovered in 1964 by Penzias and Wilson, who got a Nobel Prize for their discovery. (One of their early hypotheses for the effect they were seeing was .... pigeon crap. LOL!) The CMBR later turned out to be the most perfect blackbody known in the entire universe. The temperature of the CMBR today is very cold: approximately 2.725 Kelvin.

[some complicated curves displaying element abundance (c.f. hydrogen) for deuterium, helium-3, helium-4, depicted on the vertical axis, as sensitive functions of the density of ordinary matter (c.f. photon density), depicted on the horizontal axis]

The image below shows the COBE data compared against theory. The experimental errors on the observatinoal data points are so small that you cannot see them with this image resolution! Theory and experiment fit perfectly.

[COBE power spectrum]

Concordance Model

The best theoretical fit to current astronomical observations is a model sometimes abbreviated as ΛCDM. This includes (1) a cosmological constant (dark energy); (2) cold dark matter, which moves slowly enough to be non-relativistic, and (3) regular matter that we know and love. The physics of dark energy dominates the accelerated expansion of our large universe at this stage in its evolution. But dark energy dynamics did not dominate in the early universe.

Future of Universe?

You can check out how varying the composition of the universe today affects its future by accessing this cool NASA web tool at

[cartoon graph showing how expansion of the universe differs with time depending on how much of the total density is made of matter (dark and regular) as compared to dark energy]